Torsors in super-symmetry

Abstract

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is shown in an explicit way that, under suitable assumptions, every super-torsor arises from an ordinary torsor. Especially, the objects with affinity restriction, or namely, the affine super-torsors and the affine ordinary torsors are proved to be precisely in one-to-one correspondence. The results play substantial roles in ongoing construction of super-symmetric Picard–Vessiot theory.